{{Multiple issues| {{one source|date=March 2025}} {{more footnotes|date=March 2025}} }} A '''power-bounded element''' is an element of a topological ring whose powers are bounded. These elements are used in the theory of adic spaces.
== Definition == Let <math>A</math> be a topological ring. A subset <math>T \subset A</math> is called '''bounded''', if, for every neighbourhood <math>U</math> of zero, there exists an open neighbourhood <math>V</math> of zero such that <math>T \cdot V := \{t \cdot v \mid t \in T, v \in V\} \subset U</math> holds. An element <math>a \in A</math> is called '''power-bounded''', if the set <math>\{a^n \mid n \in \mathbb N\}</math> is bounded.<ref>Wedhorn: Def. 5.27</ref>
== Examples ==
* An element <math>x \in \mathbb R</math> is power-bounded if and only if <math>|x| \leq 1</math> hold. * More generally, if <math>A</math> is a topological commutative ring whose topology is induced by an absolute value, then an element <math>x \in A</math> is power-bounded if and only if <math>|x| \leq 1</math> holds. If the absolute value is non-Archimedean, the power-bounded elements form a subring, denoted by <math>A^{\circ}</math>. This follows from the ultrametric inequality. * The ring of power-bounded elements in <math>\mathbb Q_p</math> is <math>\mathbb Q_p^{\circ} = \mathbb Z_p</math>. * Every topological nilpotent element is power-bounded.<ref>Wedhorn: Rem. 5.28 (4)</ref>
== Literature ==
* Morel: [https://web.math.princeton.edu/~smorel/adic_notes.pdf Adic spaces]
* Wedhorn: [https://arxiv.org/pdf/1910.05934.pdf Adic spaces]
== References == <references /> Category:Topological algebra